Angles: types, measuring and angle facts

An angle is the amount of turn between two lines that meet at a point. The point is the vertex and the two lines are the arms (or sides). Angles are usually measured in degrees (°), where a full turn is 360°. Angles are named by size: acute angles are less than 90°, right angles are exactly 90°, obtuse angles are between 90° and 180°, and reflex angles are between 180° and 360°.

What is an angle?

Picture a door opening. The hinge is the vertex, the wall and the door are the arms, and the angle is how far the door has turned. An angle measures turn, not length: making the arms longer doesn't change it.

There are three common ways to name an angle:

• Three letters, with the vertex in the middle: ∠ABC, or angle ABC, is the angle at B between BA and BC.
• One letter, the vertex alone: ∠B, when there is only one angle at B.
• A small letter or Greek letter written inside the angle, such as x or θ (theta).

Types of angles

Every pair of arms makes two angles, one on each side, and they add up to 360°. Usually "the angle" means the smaller one. The bigger one is the reflex angle. So arms with a 135° angle between them also make a reflex angle of 360 − 135 = 225°.

How to measure an angle with a protractor

1. Put the centre (center in the US) of the protractor exactly on the vertex.
2. Line up the zero line of the protractor with one arm.
3. Follow the scale that starts at 0° on that arm. Most protractors have two scales, an inner and an outer one.
4. Read where the other arm crosses that scale.
5. Check it makes sense: an acute angle must read under 90°, an obtuse one over 90°.

A protractor only goes to 180°, so for a reflex angle, measure the smaller angle on the other side and take it from 360°.

Degrees and radians

In more advanced maths, angles are often measured in radians instead. A full turn is 2π radians, so 180° = π radians and one radian is about 57.3°. The unit circle guide shows where radians come from and has a table of common angles in both units.

Angle facts

These are the rules that let you work out angles without measuring. They are the same ones AQA lists for GCSE Maths, and US students meet them across middle school and high school geometry.

Each of these has its own guide with examples: vertical angles, corresponding angles, alternate interior angles, and the angles of a pentagon.

Angle pairs

Many questions name the relationship between two angles rather than the shape.

More on each: supplementary angles (with complementary angles compared) and adjacent angles.

How to find missing angles

1. Look for the shape or lines the angle belongs to: a straight line, a point, a triangle, parallel lines.
2. Pick the fact that links it to angles you know.
3. Write an equation if there is an unknown, then solve it.
4. Write the reason next to each step.

Example 1: angles in a triangle

The angles of a triangle are x, 2x and 3x. Find them.

The angles are 30°, 60° and 90°, so this is a right-angled triangle (right triangle in the US).

Example 2: an isosceles triangle

An isosceles triangle has a top angle of 40°. Its two base angles are equal, and the three angles add up to 180°, so each base angle is (180 − 40) ÷ 2 = 70°.

Example 3: the exterior angle of a triangle

A triangle has interior angles of 48° and 67°. One side is extended past the third corner. Find the exterior angle made there.

The exterior angle equals the two opposite interior angles added together: 48 + 67 = 115°. Check the long way: the third angle is 180 − 48 − 67 = 65°, and the exterior angle sits on a straight line with it, so it is 180 − 65 = 115°.

Finding angles with trigonometry

Angle facts need other angles to start from. In a right-angled triangle where you know two sides but no angles, use trigonometry instead. Pick the ratio that uses the two sides you know, then use the inverse function on your calculator.

Example 4: an angle from two sides

In a right-angled triangle, the side opposite angle θ is 5 cm and the side next to it (the adjacent) is 12 cm. Opposite and adjacent means tangent:

So θ = 22.6° to 1 decimal place. Make sure your calculator is in degree mode. Now try one yourself, then press Mark to check it in the course.

The Trigonometry course teaches SOH CAH TOA from the start, and the trigonometry calculator finds every angle of a triangle you drag into shape.

Common mistakes

• Reading the wrong protractor scale. Start from the 0° on the arm you lined up, and check the answer is sensible for the type of angle.
• Naming an angle with the vertex at the end. In ∠ABC the vertex is B, the middle letter.
• Using parallel line facts without parallel lines. Look for arrow marks or the word parallel.
• Measuring a diagram that says "not drawn to scale". Work it out from the facts instead.
• Calculator in radian mode when finding angles with trigonometry.

To practise angles mixed with other topics in a short timed paper, try today's Pre-Algebra & Algebra 1 or Geometry, Algebra 2 & SAT Math paper. In the UK, the same practice opens as GCSE Maths Foundation.

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